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On the Euler class one conjecture for fillable contact structures

Yi Liu

TL;DR

The paper proves that every oriented closed hyperbolic 3-manifold $M$ admits a finite cover with an even lattice point $w$ on the boundary of the dual Thurston norm unit ball that cannot be realized as the real Euler class of any weakly fillable contact structure, providing new counter-examples to Euler class one. The strategy connects the next-to-top Heegaard Floer term of a pseudo-Anosov mapping torus to $1$-periodic suspension-flow trajectories via Fried’s cone, and uses Agol–Wise virtual specialization together with a cyclic-cover construction to manufacture a genuine obstruction. By combining twisted Heegaard Floer theory obstructions with a virtual homology separation and a vacuum-rational-direction argument in Fried’s cone, the paper demonstrates that non-realizable Euler classes persist in finite covers. These results have implications for the Euler class one conjecture and highlight intricate interactions between foliation theory, contact geometry, and 3-manifold dynamics in the virtual setting.

Abstract

In this paper, it is proved that every oriented closed hyperbolic $3$--manifold $N$ admits some finite cover $M$ with the following property. There exists some even lattice point $w$ on the boundary of the dual Thurston norm unit ball of $M$, such that $w$ is not the real Euler class of any weakly symplectically fillable contact structure on $M$. In particular, $w$ is not the real Euler class of any transversely oriented, taut foliation on $M$. This supplies new counter-examples to Thurston's Euler class one conjecture.

On the Euler class one conjecture for fillable contact structures

TL;DR

The paper proves that every oriented closed hyperbolic 3-manifold admits a finite cover with an even lattice point on the boundary of the dual Thurston norm unit ball that cannot be realized as the real Euler class of any weakly fillable contact structure, providing new counter-examples to Euler class one. The strategy connects the next-to-top Heegaard Floer term of a pseudo-Anosov mapping torus to -periodic suspension-flow trajectories via Fried’s cone, and uses Agol–Wise virtual specialization together with a cyclic-cover construction to manufacture a genuine obstruction. By combining twisted Heegaard Floer theory obstructions with a virtual homology separation and a vacuum-rational-direction argument in Fried’s cone, the paper demonstrates that non-realizable Euler classes persist in finite covers. These results have implications for the Euler class one conjecture and highlight intricate interactions between foliation theory, contact geometry, and 3-manifold dynamics in the virtual setting.

Abstract

In this paper, it is proved that every oriented closed hyperbolic --manifold admits some finite cover with the following property. There exists some even lattice point on the boundary of the dual Thurston norm unit ball of , such that is not the real Euler class of any weakly symplectically fillable contact structure on . In particular, is not the real Euler class of any transversely oriented, taut foliation on . This supplies new counter-examples to Thurston's Euler class one conjecture.
Paper Structure (8 sections, 13 theorems, 32 equations)

This paper contains 8 sections, 13 theorems, 32 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. The dual Thurston norm unit ball
  4. Weak symplectic fillability
  5. No one-periodic trajectory, no fillable contact structure
  6. Periodic directions are virtually rare above boundary
  7. Proof of the main theorem
  8. Further discussion

Key Result

Theorem 1.2

For every oriented closed hyperbolic $3$--manifold $M$, there exists some connected finite cover $\tilde{M}$ of $M$, and some even lattice point $\tilde{w}\in H^2(\tilde{M};{\mathbb R})$ of dual Thurston norm $1$, such that $\tilde{w}$ is not the real Euler class of any weakly symplectically fillabl

Theorems & Definitions (25)

  • Conjecture 1.1: Euler Class One
  • Theorem 1.2
  • Corollary 1.3
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 3.2
  • Remark 3.3
  • Lemma 3.4
  • ...and 15 more